Kernel and Range of a Linear Mapping

[Smazmbazm]

Homework Statement

Find the kernel and range of the following linear mapping.

b) The mapping T from $P^{R} to P^{R}_{2}$ defined by

$T(p(x)) = p(2) + p(1)x + p(0)x^{2}$

The Attempt at a Solution

I'm not sure how to go about this one. Normally I would use the formula T(x) = A * v but in this case I don't know how to find A or v. Would be great if someone could point me in the right direction.

Thanks in advanced.

 

[HallsofIvy]

I assume you know that the kernel is the set mapped to 0. T(p(x))= 0 (for all x) if and only if p(2)= p(1)= p(0)= 0.

And the range is the entire set P^R₂. Do you see why?

 

[Smazmbazm]

Yes I know that the kernel is the set that is mapped to 0. I think I'm just having trouble with understanding what $T(p(x)) = p(2) + p(1)x + p(0)x^{2}$ actual means. It's a bit too general for me. What is $p(x)$? Is that saying that the power representation for $p(x)$ is $p(2) + p(1)x + p(0)x^{2}$? Or the mapping of the power series $p(x)$ from $P^{R}$ to $P^{R}_{2}$ results in $p(2) + p(1)x + p(0)x{2}$

I think I understand why the range is the entire set $P^{R}_{2}$, because if the kernel only contains the zero vector then the range must contain everything else?